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Discrete o-minimal structures

1987
*
Annals of Pure and Applied Logic
*

We assume that the reader is familiar 1 It will be shown in a forthcoming paper by Pillay and Steinhorm that any

doi:10.1016/0168-0072(87)90004-2
fatcat:rfoqwoxld5grhfoth2juynuyhi
*o*-*minimal**structure*has a strongly*o*-*minimal*theory. ... if every model of T is*o*-*minimal*. ... of n-tuples in*discretely*ordered*structures*with strongly*o*-*minimal*theories. ...##
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Compression Schemes, Stable Definable Families, and o-Minimal Structures

2009
*
Discrete & Computational Geometry
*

As a consequence, the combinatorial complexity (or density) of any * hujohnson@jjay.cuny.edu † Partially supported by NSF grant DMS-0600217. definable family in a

doi:10.1007/s00454-009-9201-3
fatcat:44jjhnaajren5hxl7ijffrieie
*structure*with a*o*-*minimal*theory is bounded ... We show that any family of sets uniformly definable in an ominimal*structure*has an extended compression scheme of size equal to the number of parameters in the defining formula. ... Lemma 4. 7 . 7 For any*o*-*minimal**structure*M and for any formula θ(x; y,z) Theorem 4. 9 . 9 Let M be any*o*-*minimal**structure*. ...##
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Locally o-minimal structures with tame topological properties
[article]

2021
*
arXiv
*
pre-print

We consider locally

arXiv:2006.13481v3
fatcat:wghfya4plvhyhghzopgthqjxum
*o*-*minimal**structures*possessing tame topological properties shared by models of DCTC and uniformly locally*o*-*minimal*expansions of the second kind of densely linearly ordered abelian ... We derive basic properties of dimension of a set definable in the*structures*including the addition property, which is the dimension equality for definable maps whose fibers are equi-dimensional. ... Acknowledgment In the initial draft, the author considers*structures*simultaneously satisfying the property (a) in Definition 1.1 and the properties (b) through (d) in Definition 2.6, ignoring their dependence ...##
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O-MINIMALISM

2014
*
Journal of Symbolic Logic (JSL)
*

This paper is devoted to

doi:10.1017/jsl.2013.14
fatcat:hcrqbgyihfej3kt424wfrgeyuu
*o*-*minimalism*, the study of the first-order properties of*o*-*minimal**structures*. ... To any pseudo-*o*-*minimal**structure*, we can associate its Grothendieck ring, which in the non-*o*-*minimal*case is a nontrivial invariant. ... Pseudo-*o*-*minimal**structures*§10.*O*-*minimalism*. ...##
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Locally o-minimal structures and structures with locally o-minimal open core

2013
*
Annals of Pure and Applied Logic
*

We give a characterisation of

doi:10.1016/j.apal.2012.10.002
fatcat:fmwztdzvvva35prhg3qunus77i
*structures*with locally*o*-*minimal*open core, and we show that dense elementary pairs of locally*o*-*minimal**structures*have locally*o*-*minimal*open core. ... We study first-order expansions of ordered fields that are definably complete, and moreover either are locally*o*-*minimal*, or have a locally*o*-*minimal*open core. ... One advantage of locally*o*-*minimal**structures*over*o*-*minimal*ones is that they form an elementary class: that is, an ultraproduct of locally*o*-*minimal**structures*is locally*o*-*minimal*. ...##
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The Non-Axiomatizability of O-Minimality
[article]

2012
*
arXiv
*
pre-print

Call an L-

arXiv:1203.2715v2
fatcat:xjjhehrksndedflcxecq4nrqii
*structure*R pseudo-*o*-*minimal*if it is (elementarily equivalent to) an ultraproduct of*o*-*minimal**structures*. ... This answers negatively a question raised by Schoutens, and shows that the theory consisting of those L-sentences true in all*o*-*minimal*L-*structures*, called the theory of*o*-*minimality*(for L), is not recursively ... Thus, in the bare field language, pseudoo-*minimal**structures*are*o*-*minimal*. ...##
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A Penalty Approach for Nonlinear Optimization with Discrete Design Variables
[chapter]

1989
*
Discretization Methods and Structural Optimization — Procedures and Applications
*

Although numerous algorithms for

doi:10.1007/978-3-642-83707-4_42
fatcat:hltnvbqhjvhmlaqdzzpgw67rju
*structural*optimization problems have been developed, relatively little effort has been made for optimum*structural*design incorporating design variables with*discrete*... This means only t h e*o*r i g i n a l*o*b j e c t i v e function and c*o*n s t r a i n t penalty terms t a k e p a r t i n t h e*minimization*process. ...##
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O-minimalism
[article]

2011
*
arXiv
*
pre-print

An ordered

arXiv:1106.1196v1
fatcat:ik5lb72e6nhgtm3r746en3m7nq
*structure*is called*o*-minimalistic if it has all the first-order features of an*o*-*minimal**structure*. ... To any*o*-minimalistic*structure*, we can associate its Grothendieck ring, which in the non-*o*-*minimal*case is a non-trivial invariant. ... An*o*-minimalistic*structure*with*o*-*minimal*open core is itself*o*-*minimal*. Proof. ...##
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Expansions of o-minimal structures by fast sequences

2005
*
Journal of Symbolic Logic (JSL)
*

Let ℝ be an

doi:10.2178/jsl/1120224720
fatcat:whndm5mvhrai7hutqdpidha4tm
*o*-*minimal*expansion of (ℝ, <, +) and (ϕ k ) kЄℕ be a sequence of positive real numbers such that limt k →+∞ f (ϕ k )/ϕ k +1 = 0 for every f: ℝ → ℝ definable in ℜ (Such sequences always exist ... Then (ℜ, (S)) is d-*minimal*. where S ranges over all subsets of cartesian powers of the range of ϕ. ... Proofs of Theorems 2 and 3 The reader is now assumed to be familiar with basic properties of*o*-*minimal**structures*, e.g., [3, Chapters 1 and 3]. ...##
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Definable sets in ordered structures. III

1988
*
Transactions of the American Mathematical Society
*

Thus, with the results of §1,

doi:10.1090/s0002-9947-1988-0943306-9
fatcat:wuswpwwrvbfyxgwgbbqlrukb5q
*discrete**o*-*minimal**structures*lose their interest. ... We show that any*o*-*minimal**structure*has a strongly*o*-*minimal*theory. 0. Introduction. In this paper we prove that an arbitrary*o*-*minimal**structure*M is strongly*o*-*minimal*. ... Mixed*o*-*minimal**structures*. Here we will show that an arbitrary ominimal*structure*M is strongly*o*-*minimal*. ...##
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Definable Sets in Ordered Structures. III

1988
*
Transactions of the American Mathematical Society
*

Thus, with the results of §1,

doi:10.2307/2000920
fatcat:sdzjx6j6ubc4fbyh3fjbal76ga
*discrete**o*-*minimal**structures*lose their interest. ... We show that any*o*-*minimal**structure*has a strongly*o*-*minimal*theory. 0. Introduction. In this paper we prove that an arbitrary*o*-*minimal**structure*M is strongly*o*-*minimal*. ... Mixed*o*-*minimal**structures*. Here we will show that an arbitrary ominimal*structure*M is strongly*o*-*minimal*. ...##
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On the expressiveness and decidability of o-minimal hybrid systems

2005
*
Journal of Complexity
*

We show that this can be done for a quite general class of hybrid systems defined on

doi:10.1016/j.jco.2004.09.003
fatcat:24vweszpuzgrbj5sd3g5iei6ia
*o*-*minimal**structures*. ... In the case of*o*-*minimal*hybrid systems, the continuous and*discrete*components can be decoupled, and hence, the problem reduces in building a finite symbolic dynamical system for the continuous dynamics ... There exists many more interesting*o*-*minimal**structures*. The main result we use on*o*-*minimal**structures*is (see [vdD98, Corollary 3 .6, p. 60]): Theorem 2.13 (Uniform Finiteness). ...##
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SHAPE DESIGN OF MEMBRANE STRUCTURE USING GEOMETRIC INVARIANTS OF DISCRETE SURFACE

2021
*
Journal of Structural and Construction Engineering (Transactions of AIJ)
*

The surface is

doi:10.3130/aijs.86.772
fatcat:nspiaepgqnf27cwh6lbnywjcj4
*discretized*into triangular faces. ... Optimization procedures are proposed for design of frame-supported membrane*structures*. A developable, a*minimal*, and their intermediate surfaces are chosen as the target surfaces. ... Suris:*Discrete*Differential Geometry: Integrable*Structure*, American Mathematical Society, 2008. 16) J. M. Sullivan: Curvatures of smooth and*discrete*surfaces,*Discrete*Differential Geometry. ...##
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Locally o-minimal open core
[article]

2021
*
arXiv
*
pre-print

We demonstrate that the open core of a definably complete expansion of a densely linearly ordered abelian group is locally

arXiv:2111.08197v1
fatcat:tumlgb3x4nfqvc6fkdwoqebduq
*o*-*minimal*if and only if any definable closed subset of R is either*discrete*or ... Here, the notation R denotes the universe of the original*structure*. ... Definably complete uniformly locally*o*-*minimal**structures*of the second kind satisfy the additional condition which general locally*o*-*minimal**structures*do not satisfy. ...##
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Dp-minimal expansions of discrete ordered abelian groups
[article]

2020
*
arXiv
*
pre-print

If Z is a dp-

arXiv:1903.06222v2
fatcat:d53yih5fpndwhjnmk6pkr4n5wq
*minimal*expansion of a*discrete*ordered abelian group (Z,<,+) and Z does not admit a nontrivial definable convex subgroup then Z is interdefinable with (Z,<,+) and (Z,<,+) is elementarily ... Many*structures*whose definable sets are "tame topological" objects are dp-*minimal*. Dp-*minimal*theories include*o*-*minimal*, Cminimal, P-*minimal*, strongly*minimal*theories, and Presburger arithmetic. ... Throughout this paper (Z, <, +) is a*discrete*ordered abelian group, 1 Z is the*minimal*positive element of Z,*O*∶= {k1 Z ∶ k ∈ Z}, and Z is an expansion of (Z, <, +). ...
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